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Question
Evaluate sin
\[\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right)\]
Solution
We know that
\[\cos^{- 1} x = 2 \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}}\]
\[ \tan^{- 1} x = \sin^{- 1} \frac{x}{\sqrt{1 + x^2}}\]
\[\therefore \sin\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right) = \sin\left( \frac{1}{2}2 \tan^{- 1} \sqrt{\frac{1 - \frac{4}{5}}{1 + \frac{4}{5}}} \right)\]
\[ = \sin\left( \tan^{- 1} \sqrt{\frac{\frac{1}{5}}{\frac{9}{5}}} \right)\]
\[ = \sin\left( \tan^{- 1} \frac{1}{3} \right)\]
\[ = \sin\left\{ \sin^{- 1} \left( \frac{\frac{1}{3}}{\sqrt{1 + \frac{1}{9}}} \right) \right\}\]
\[ = \sin\left( \sin^{- 1} \frac{1}{\sqrt{10}} \right)\]
\[ = \frac{1}{\sqrt{10}} \left[ \because \sin\left( \sin^{- 1} x \right) = x \right]\]
∴ \[\sin\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right) = \frac{1}{\sqrt{10}}\]
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